Mortgage Payments, etc.


BUSI 721
Jones Graduate School of Business
Rice University

Kerry Back

Questions about loans

  • I know how much I want to borrow and for how many years and what the rate will be. What will my payment be?
  • I know how much I can afford to pay and for how many years the loan will be and what the rate will be. How much can I borrow?
  • I know how much I want to borrow and what I can afford to pay and for how many years the loan will be. What rate do I need to get?

  • Suppose we borrow for \(T\) years at an annual rate of \(r\) with monthly payments.
  • Our monthly rate will be \(r_m = r/12\).
  • Each month, we incur interest equal to \(r_m\) times the beginning-of-month balance.
  • Our payment goes first to interest, then the remainder to principal, reducing the outstanding balance.
  • Next month’s interest charge is smaller, so more goes towards principal.
  • The last payment is equal to the beginning-of-month balance plus interest on the balance.

  • Call the amount borrowed (principal of the loan) \(B\) and the payment \(P\).
  • What we owe just after the first payment is \((1+r_m)B-P\).
  • What we owe just after the second payment is

\[(1+r_m)((1+r_m)B - P) - P\]

  • This is \((1+r_m)^2 B - (1+r_m)P - P\), which is the future value of the amount borrowed minus the future value of the payments (as of the end of the second month).

  • Because the loan is eventually paid off, the future value of the amount borrowed equals the future value of the payments as of the end of the last month.

Future values

  • The balance after the last payment is zero.
  • So, for future values as of the loan maturity,

\[0 = \text{FV(Principal)} - \text{FV(Payments)}\]

Equivalently,

\[\text{FV(Principal)} = \text{FV(Payments)}\]

Present values

  • Because future values are equal, present values are also equal.
  • The present value of the amount borrowed is just the amount (prinicpal).
  • So loan terms are related by Principal = PV of payments.
  • The PV of payments is

\[P \times \left(\frac{1}{1+r_m} + \frac{1}{(1+r_m)^2} + \cdots + \frac{1}{(1+r_m)^{12 \times T}}\right)\]

Annuity factor

  • The sum of PV factors is called the annuity factor AF.
  • So, Principal = Payment \(\times\) AF.
  • There is a simpler formula for the annuity factor (sum of a finite number of terms of a geometric series):

\[\text{AF} = \frac{1}{r_m}\left(1 - \frac{1}{(1+r_m)^{12 \times T}}\right)\]

  • It is not necessary to remember the AF formula.

Balloon payments

  • A loan might have a balloon payment, meaning a lump sum payment at the end.
  • Loan terms are related by Principal = PV of monthly payments + PV of balloon
  • PV of balloon is

\[\frac{\text{Balloon}}{(1+r_m)^{12 \times T}}\]

Numpy financial

  • You may need to do
    pip install numpy-financial
  • Convention is that payments, including balloon, are negative.
  • Principal received is positive. It is called pv.
  • Ballon is called fv.
  • Functions are rate, pmt, pv, fv.
  • Same functions are in Excel.

Example 

import numpy_financial as npf
 
nper = 30 * 12       # 30 year loan
principal = 400000   # borrow $400,000
pmt = -1500          # pay $1,500 per month
rate = 0.04 / 12     # annual rate is 4%
balloon = 0          # no balloon

We’ll calculate each of the last four, one at a time, given the other inputs.

# Question 1: what will my payment be?
PMT = npf.pmt(rate=rate, nper=nper, pv=principal, fv=balloon)
 
# Question 2: how much can I borrow?
PRINCIPAL = npf.pv(rate=rate, nper=nper, pmt=pmt, fv=balloon)

# Question 3: what will my rate be?
RATE = npf.rate(nper=nper, pv=principal, pmt=pmt, fv=balloon)

# And, how much of a balloon do I need?
BALLOON = npf.fv(rate=rate, nper=nper, pmt=pmt, pv=principal)

If BALLOON>0, you overpaid the bank and should get money back.